Theorem sstpr 3519

Description: The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)

Assertion

Ref	Expression
sstpr	⊢ ((((A = ∅ ∨ A = {B}) ∨ (A = {𝐶} ∨ A = {B, 𝐶})) ∨ ((A = {𝐷} ∨ A = {B, 𝐷}) ∨ (A = {𝐶, 𝐷} ∨ A = {B, 𝐶, 𝐷}))) → A ⊆ {B, 𝐶, 𝐷})

Proof of Theorem sstpr

Step	Hyp	Ref	Expression
1		ssprr 3518	. . 3 ⊢ (((A = ∅ ∨ A = {B}) ∨ (A = {𝐶} ∨ A = {B, 𝐶})) → A ⊆ {B, 𝐶})
2		prsstp12 3508	. . 3 ⊢ {B, 𝐶} ⊆ {B, 𝐶, 𝐷}
3	1, 2	syl6ss 2951	. 2 ⊢ (((A = ∅ ∨ A = {B}) ∨ (A = {𝐶} ∨ A = {B, 𝐶})) → A ⊆ {B, 𝐶, 𝐷})
4		snsstp3 3507	. . . . 5 ⊢ {𝐷} ⊆ {B, 𝐶, 𝐷}
5		sseq1 2960	. . . . 5 ⊢ (A = {𝐷} → (A ⊆ {B, 𝐶, 𝐷} ↔ {𝐷} ⊆ {B, 𝐶, 𝐷}))
6	4, 5	mpbiri 157	. . . 4 ⊢ (A = {𝐷} → A ⊆ {B, 𝐶, 𝐷})
7		prsstp13 3509	. . . . 5 ⊢ {B, 𝐷} ⊆ {B, 𝐶, 𝐷}
8		sseq1 2960	. . . . 5 ⊢ (A = {B, 𝐷} → (A ⊆ {B, 𝐶, 𝐷} ↔ {B, 𝐷} ⊆ {B, 𝐶, 𝐷}))
9	7, 8	mpbiri 157	. . . 4 ⊢ (A = {B, 𝐷} → A ⊆ {B, 𝐶, 𝐷})
10	6, 9	jaoi 635	. . 3 ⊢ ((A = {𝐷} ∨ A = {B, 𝐷}) → A ⊆ {B, 𝐶, 𝐷})
11		prsstp23 3510	. . . . 5 ⊢ {𝐶, 𝐷} ⊆ {B, 𝐶, 𝐷}
12		sseq1 2960	. . . . 5 ⊢ (A = {𝐶, 𝐷} → (A ⊆ {B, 𝐶, 𝐷} ↔ {𝐶, 𝐷} ⊆ {B, 𝐶, 𝐷}))
13	11, 12	mpbiri 157	. . . 4 ⊢ (A = {𝐶, 𝐷} → A ⊆ {B, 𝐶, 𝐷})
14		eqimss 2991	. . . 4 ⊢ (A = {B, 𝐶, 𝐷} → A ⊆ {B, 𝐶, 𝐷})
15	13, 14	jaoi 635	. . 3 ⊢ ((A = {𝐶, 𝐷} ∨ A = {B, 𝐶, 𝐷}) → A ⊆ {B, 𝐶, 𝐷})
16	10, 15	jaoi 635	. 2 ⊢ (((A = {𝐷} ∨ A = {B, 𝐷}) ∨ (A = {𝐶, 𝐷} ∨ A = {B, 𝐶, 𝐷})) → A ⊆ {B, 𝐶, 𝐷})
17	3, 16	jaoi 635	1 ⊢ ((((A = ∅ ∨ A = {B}) ∨ (A = {𝐶} ∨ A = {B, 𝐶})) ∨ ((A = {𝐷} ∨ A = {B, 𝐷}) ∨ (A = {𝐶, 𝐷} ∨ A = {B, 𝐶, 𝐷}))) → A ⊆ {B, 𝐶, 𝐷})

Syntax hints:   → wi 4   ∨ wo 628   = wceq 1242   ⊆ wss 2911  ∅c0 3218  {csn 3367  {cpr 3368  {ctp 3369

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3or 885  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-tp 3375

This theorem is referenced by:  pwtpss  3568